EP2149221A2 - Method of authentication of an entity by a verifying entity - Google Patents

Abstract

Method of authentication of an entity by a verifying entity, said entities sharing a pair of secret keys X and Y. According to the invention, said secret keys X and Y are n ´ m binary matrices (n,m >l), said method comprising steps repeated r times (r ≥ l) consisting: - for the entity (1) to be authenticated and the verifying entity (2), in exchanging binary vectors a and b of, n bits respectively, drawn randomly by the verifying entity (2) and the entity (1) to be authenticated; and, for the entity (1) to be authenticated, in randomly drawing a binary noise vector c of m bits, each of said m bits being equal to 1 with a probability η of less than 1/2, and in calculating and transmitting to the verifying entity (2) a response vector z of m bits equal to, - for the verifying entity, in calculating the Hamming weight (220') of an error vector, then, for the verifying entity, in accepting (240') the authentication if the Hamming weight of the r error vectors e satisfy a comparison relation (230') with a parameter (T) dependent on the probability η. Application to cryptographic protocols for authenticating electronic chips at very low cost.

Description

 METHOD FOR AUTHENTICATING AN ENTITY BY AN ENTITY

Auditor

The present invention relates to a method of authenticating an entity to be authenticated with a checking entity.

The invention finds a particularly advantageous application in the field of cryptographic authentication protocols of microchips at very low cost, with or without contact, including radio frequency tags RFID ("Radio Frequency IDentification").

Low-cost electronic chips, such as RFID for example, are used in many applications such as labeling and tracing of objects (drugs, library books, etc.), or the production and verification of tickets or electronic tickets, such as tickets.

Whatever the application considered, it is necessary to prevent fraud that may occur on the basis of the falsification of chips, in particular their copying, or cloning, or the replay of the data they transmit. In order to protect the applications against such attacks, it is imperative to authenticate the chips during their interactions with a reader.

However, any authentication protocol between an entity to be authenticated, such as a low-cost chip, and a reader acting as a verifying entity, must take into account the extreme limitation of the calculation resources of this type of chip, which are most often wired logic.

Recently, it has been proposed (A. Juels and SA Weis, "Authenticating Pervasive Devices with Human Protocols", in V. Shoup, Editor, Advances in Cryptology-Crypto 05, Lectures Notes in Computer Science, Vol 3126, pp. 293 -308, Springer Verlag) a symmetric authentication protocol specifically designed to meet the needs of RFID chips. This protocol is known as HB + (Hopper-Blum). FIG. 1 represents the data exchanges performed during the HB + protocol between the entity to be authenticated and the audit entity.

As can be seen in this figure, the entity to authenticate, a chip

RFID for example, and the checking entity, a reader of the chip, share a pair of secret keys x and y constituted by binary vectors of n bits.

These secret keys are stored in storage means of the chip and the reader referenced respectively 10 and 20.

The HB + protocol is run on successive cycles. At each turn, the chip randomly draws (block 100) and transmits (1) to the reader a binary vector b of n bits. Similarly, the reader randomly draws (block 200) and transmits (2) to the chip an a binary vector of n bits. The drawing of the vectors b and a is done according to a uniform probability law.

The chip then responds to the challenge launched by the reader by calculating (block

120) and by transmitting to it (3) a noisy response z = a »x ®b» y ® v, where • represents the scalar product operation modulo 2 and ® the addition modulo 2. v is a noise bit drawn at leaves by the chip (block 1 10); it takes the value 1 with a probability η <1/2 and the value 0 with a probability (I - / 7).

The reader rejects the current round (block 210) if the response z received does not satisfy the relation z = a »x ®b» y; in this case, a counter of rejected rpm numbers nbr is incremented by one unit (block 220). At the end of the rounds, counted by a counter of the number of rounds nbt (block 250), the authentication is accepted (block 240) if and only if the number of rejected rounds nbr of the counter is lower than a given threshold t (block 230). The value of t is of course a function of the probability η; a simple value of t is for example t = r x η.

Although the exchanges of the proposed HB + protocol are structured in revolutions of three passes, it is possible to reduce to a three-pass exchange by calculating and transmitting at once r values of b, a and z.

The advantage of the HB + protocol lies in the great simplicity of the authentication calculations.

In addition, it draws its robustness from the difficulty of the problem LNP ("Low Parity with Noise") to find a solution to a noisy linear system. Finally, the HB + protocol benefits, by comparison with the HB protocol historically prior and differing in that the noise response did not include a term b * y, because of the masking effect induced by the binary vector b coupled to the secret key y; indeed, the HB protocol was sensitive to attacks consisting of the opponent sending constant challenges and listening to the reader's responses; the most frequent answer being a »x, and a being known, one could at first obtain a» x for a sufficient number of values of a, and in a second time deduce x by resolution of a linear system.

However, the HB + protocol has drawbacks that prohibit it from being used effectively in practice.

A first disadvantage is that even if, as we have seen, it resists some active attacks on a, this protocol is nevertheless vulnerable to other attacks encountered when an adversary has access to results in terms of success / failures of several successive authentications.

Such an attack consists in intercepting the challenge when transmitting the reader to the chip and successively modifying the bits. If, for example, the first bit of a is modified, it is understood that if the result is not changed as a result of this modification, it can be concluded that the first bit of the secret vector x is probably 0. Conversely, if the result is changed, the first bit of x is probably equal to 1. To obtain all of the n bits of x, it suffices to modify the second bit of a to know the second bit of x, and so on until not .

A second disadvantage of the HB + protocol is that it results in an excessively high number of false alerts, a false alarm being defined as the rejection of the authentication of legitimate chips. Thus, for example, with the values "= 224 bits, / 7 = 0.25, r = 100 revolutions and t = η xr = 25, the false alarm rate is 45%, a value that is totally unacceptable. The false-positive rate, that is, the success of authentication for random chips, is close to 3 ^• 10 ^{~ 7} .

If instead of taking for the expected value η xr = 25, we take a higher value, for example 35, the false alarm rate goes down to 1%, which remains unacceptable, but the rate of false positives increases to about 1.7 - 10 ^"3 .

Finally, a third disadvantage of HB + is the excessive complexity that it induces in the communications between the chip and the reader. With the same figures as before, it can be seen that it is necessary to exchange 44900 bits with each authentication, ie, at each of the 100 turns, 224 bits for b, 224 bits for a, and one bit for the result z.

It can be seen that, even with a bit rate of 10000 bits / s, it takes more than four seconds for the reader to authenticate a chip, which remains a prohibitive value for the ergonomics of the system, not to mention the power supply problems of the chip. who as a result.

The invention therefore relates to a method of authenticating an entity with a checking entity, said entities sharing a pair of secret keys X and Y. Said method is remarkable in that said secret keys X and Y are binary matrices n x m (n, m> 1), and that it comprises repeated steps r times (r> 1) consisting of:

for the entity to be authenticated and the auditing entity, to exchange binary vectors a and b of n bits randomly selected by the verifier entity and the entity to be authenticated, and, for the entity to be authenticated, to randomly selecting a binary noise vector c of m bits, each of said m bits being equal to 1 with a probability η less than 1/2, and calculating and transmitting to the checking entity a response vector z of m bits equal to at z = aX ® bY ® c,

- for the auditing entity, to calculate the Hamming weight of an error vector e = z ® aX ®bY, then, for the auditing entity, to accept the authentication if the Hamming weights of the r vectors d error e verify a relation of comparison to a parameter function of the probability η.

Thus, it can be seen that the method according to the invention offers, compared to the HB + protocol, a better resistance to attacks consisting in modifying the bits of the challenge a in order to reconstitute the secret X. Indeed, if the first bit of a is modified, this modification affects the products of this bit with the first bits of the m columns of X, which also affects the m bits of the product aX and therefore the response z as a whole. Therefore, it is not possible to deduce from the observation of the effect of a bit modification of a on the authentication result any information on the secret X since several of the m bits of z can be modified without we can know their number and their position, whereas in the case of the HB + protocol any modification of a bit of a directly affects the response z, this one consisting of only one bit.

Regarding the performance of the method, object of the invention, it should be noted that, the response z at each round on m bits, everything happens in substance as if we performed m turns in one. As a result, in a first extreme situation, it is possible to reduce by a factor of the order of m the number of revolutions, and therefore in practice limit the number of revolutions to one, which makes it possible to maintain the performances of the HB + protocol in false alarm rate, but reducing the number of bits exchanged from r (2n + \) to (2n + m), ie from 44900 bits to 576 bits, with "= 224 and m = 128 which represents a considerable gain. It is understood in this example the interest of the invention to allow to limit the number r of revolutions, which is impossible to envisage in the case of the HB + protocol.

In a second extreme situation, the number of turns is the same. In this case, the number of data exchanged increases slightly, but against the false alarm rate becomes insignificant.

Of course, a realistic situation will be chosen between these two extreme situations with both a reduction in the rate of false alarms and the number of bits exchanged between the chip and the reader. Be that as it may, it is clear that the present invention offers better performance than the HB + protocol in terms of false alarm rates and the amount of information to be exchanged between the two entities concerned.

According to a particular embodiment of the invention, said matrices X and Y are matrices of Toeplitz. It will be seen in detail later that this advantageous arrangement makes it possible to limit the storage capacity of chips and readers to (n + m - 1) instead of nxm in the case of selected dies. any. Another advantage is to simplify the calculation of the products άX and bY.

The invention also relates to an entity intended to be authenticated by a checking entity, said entities sharing a pair of secret keys X and Y, remarkable in that said entity to be authenticated comprises means for storing secret keys X and Y constituted by binary matrices nxm (n, m> 1), communication means with the checking entity, and calculation means adapted to perform r times (r ≥ 1) the steps of: - drawing lots and transmitting to the entity checking a binary vector b of n bits,

to receive from the auditing entity a bit vector a of n bits,

randomly drawing a binary noise vector c of m bits, each of said m bits being equal to 1 with a probability η less than 1/2, and calculating and transmitting to the checking entity a response vector z of m bits equal to z = aX @ bY @ c.

The invention further relates to a computer program comprising program instructions for implementing the steps performed by said entity to authenticate when said program is executed by a computer forming part of said computing means of the entity to be authenticated.

The invention also relates to a verification entity sharing a pair of secret keys X and F with an entity to be authenticated, which is remarkable in that the said checking entity comprises means for storing secret keys X and Y constituted by nx bit matrices. m (n, m> l), communication means with the entity to be authenticated, and calculation means able to perform r times (r ≥ 1) the steps of:

to receive from the entity to authenticate a binary vector b of n bits,

randomly and transmit to the entity to authenticate an n-bit binary vector, to receive from the entity to authenticate an m bit response vector,

- to calculate the Hamming weight of an error vector e = z ® aX ® bY, and to accept the authentication if the Hamming weights of the r error vectors e verify a comparison relation to a parameter (T, t) function of a predetermined probability η.

Finally, the invention relates to a computer program comprising program instructions for carrying out the steps performed by said checking entity when said program is executed by a computer forming part of said checking entity calculation means.

The description, given by way of non-limiting example, which follows with reference to the accompanying drawings will make it clear what the invention is and how it can be achieved. FIG. 2 represents the exchanges between the entity to be authenticated and the audit entity during the process according to the invention.

FIG. 3 is a diagram of an entity to be authenticated according to the method of FIG. 2.

FIG. 4 is a diagram of a verifying entity responsible for authenticating the entity of FIG. 3 according to the method of FIG. 2.

FIG. 2 illustrates an authentication method enabling a checking entity, a chip reader with or without contact, for example, to verify the identity of an entity to be authenticated which, in this example, may be an RFID chip . The method described in FIG. 2 is a so-called symmetrical method where the two entities, chip and reader, share the same secret keys. These, designated X and Y, are binary matrices n x m (n, m> 1) with n rows and m columns. The secret keys X and Y are stored in storage means of the chip and the reader referenced respectively 10 and 20 in FIG. 2, as well as in FIGS. 3 and 4.

The process according to the invention is structured in repeated steps r times (r> 1). The term "r times" means that the exchanges between the chip and the reader can be carried out sequentially in revolutions of three passes, as indicated in FIG. 2, where the number of revolutions nbt is incremented by 1 at each turn by a counter (block 250), or in parallel over three passes, each pass comprising the transmission of r data from one entity to another. In the example of Figure 2, chip 1 draws at each turn (block

100) and transmits (1) to the reader 2 a single binary vector b of n bits. The reader 2 then transmits (2) to the chip 1 a challenge a (block 200) which is also a single-ended binary vector of n bits drawn at random. The draw of the binary vectors b and a is performed according to a uniform distribution of bits 0 and 1.

In response to the challenge a, the chip 1 transmits (3) to the reader 2 a m bit binary vector z of m bits equal to the sum modulo 2 z = aX ® bY ® c (block 120 '), where c is a one-bit binary vector m bit noise, randomly drawn (block 1 10 ') by the chip 1 according to a probability law ensuring that each bit of c is equal to 1 with a probability equal to or less than or equal to a parameter η less than 1/2. To do this, each bit of the noise vector c can be drawn independently, according to a Bemoulli law of parameter η <1/2. The noise vector c can also be drawn from among the set of m bit vectors whose sum of bits, or Hamming weight, is at most equal to or equal to the value η xm with η <\ l 2. Good Of course, the noise vector c can be drawn by the chip at the same time as it draws the binary vector b which will be recalled that it serves as a masking to the active attacks on the vector a.

At each turn, the reader 2 calculates (block 210 ') an error vector e of m bits equal to e = z ® aX ® bY where z is the response vector sent by the chip 1, as well as the weight of Hamming PH (e) (block 220 ') of the error vector e thus obtained.

At the end of the rounds, acceptance or rejection of the authentication of the chip 1 by the reader 2 is determined from the Hamming weights PH (e) of the error vectors e obtained at each turn and from their comparison to a parameter function of the probability η.

Several strategies are then possible.

A first strategy, represented in FIG. 2, consists in accepting the authentication (block 240 ') if and only if the sum S of the Hamming weights of the r error vectors e (block 221') is less than a threshold T. given (block 230 '), equal for example to r (η + ε) m where ε is a margin less than 1/2, possibly zero. A second strategy is to accept the authentication if and only if the Hamming weight of the error vector e obtained at each turn is less than a threshold t.

Finally, a third strategy is to accept the authentication if and only if the Hamming weight of the error vector e obtained at each turn is equal to a value t.

In these two latter cases, the parameter t is (η + ε) m where ε is a margin less than 1/2, possibly zero.

By taking r = 1, n = 256, m = 128, η = 0.25 and with a random draw of the noise vector c among the bit vectors of length 128 and of Hamming weight 32, ie η xm, Authentication of the chip will be accepted according to the third strategy above if and only if the weight of the error vector e is at each turn exactly equal to 32, this with ε = 0.

In this example, we can see that the total length of the exchanges is only 640 bits, ie (2n + m). On the other hand, one can observe that the false alarm rate is strictly zero and the false positive rate for an attack of trying a random value z is close to 10 ^8, which is quite acceptable in practice .

In FIG. 2, the sequence of exchanges between chip 1 and reader 2 is: sending b to the reader, sending a to the chip, drawing c by the chip and sending z to the reader. However, it should be noted that another sequence could also be used, namely: sending a to the chip, drawing b and c by the chip and sending b and z to the reader. This last sequence has the advantage of reducing the number of exchanges. According to one embodiment which makes it possible to greatly reduce the amount of memory necessary to store the X and Y matrices, as well as the complexity of the calculations to be performed by the chip and the audit entity, each of the matrices X and Y can be selected at inside a strict subset of the set of matrices nxm and described in the chip using a number of bits strictly less than nxm. Thus, for example, the amount of memory needed to store each matrix can be reduced to only (n + m -1) when X and Y are Toeplitz matrices, namely matrices with constant coefficients along the diagonals, and whose set of coefficients is entirely determined by the coefficients of the first row and the first column. If X is a matrix of Toeplitz and if X _1} denotes the coefficient of the ith row and the jth column, x _h] is equal to v, ₊ u if * ^is greater than or equal to j, and χ _{U] → ι} otherwise.

The following embodiment makes it possible to very efficiently perform bit by bit the product of a binary vector, for example, and a Toeplitz matrix, X for example, described by means of the (n + m - 1) coefficients. of its first row and its first column, and using two registers of m bits, one to calculate the current row of the matrix and the other, initialized to 0, to accumulate the partial results of the vector-matrix product. The first register is initialized using the first line of X, then each of the bits of the vector a is treated in the following way: if the current bit of a is equal to 1, the value of the current line of X is combined by "or" exclusive bitwise with the current value of the register of accumulation of partial results. Otherwise, the current value of this register is not changed. In both cases, as long as the current line is not the last line of the matrix X, the register containing the current row of this matrix is updated by rotating the contents of this register one bit towards the right, followed by the copy in the left cell of this register of the coefficient of the first column corresponding to the new current line.

As shown in FIG. 3, the chip 1 intended to be authenticated by the reader 2, these two entities sharing a pair of secret keys X and Y, comprises means 10 for storing the secret keys X and Y constituted by binary matrices nxm (n, m> l), means 12 of communication with the reader 2, and means 1 1 of calculation able to perform r times (r ≥ 1) the steps of, according to the method described with reference to Figure 2:

randomly draw and transmit to reader 2 a binary vector b of n bits, to receive from reader 2 an n bit binary vector,

randomly drawing a binary noise vector c of m bits, each of said m bits being equal to 1 with a probability η less than 1/2, and calculating and transmitting to reader 2 a response vector z of m bits equal to z = aX ® bY ® c.

Similarly, we can see in Figure 4 a reader 2 responsible for authenticating a chip 1, comprising means 20 for storing the secret keys X and Y constituted by binary matrices nxm (n, m> l), means 22 communication with the chip 1 to authenticate, and calculation means 21 capable of performing r times (r ≥ 1) the steps of, according to the method described with reference to FIG.

to receive from the chip 1 to authenticate a binary vector b of n bits, to draw lots and to transmit to the chip 1 a binary vector with n bits,

to receive from chip 1 a response vector z with m bits,

calculating the Hamming weight of an error vector e = z® aX® bY, and accepting the authentication if the Hamming weights of the r error vectors e satisfy a comparison relation to a parameter that is a function of the probability η. In particular, authentication is accepted if the sum of the Hamming weights of the error vectors e obtained on the revolutions is less than a parameter equal to a threshold T, as has been described in detail above.

Claims

1. A method of authenticating an entity (1) with a checking entity (2), said entities sharing a pair of secret keys X and Y, characterized in that said secret keys X and Y are binary matrices nxm (n, m> l), said method comprising repeated steps r times (r ≥ 1) consisting of: for the entity (1) to be authenticated and the verifier entity (2), to exchange binary vectors a and b of n bits respectively randomly selected by the checking entity (2) and the entity (1) to authenticating, and, for the entity (1) to be authenticated, randomly drawing a binary noise vector c of m bits, each of said m bits being equal to 1 with a probability η less than 1/2, and calculating and transmitting to the auditing entity (2) an answer vector z of m bits equal to z = aX @ bY @ c, - for the auditing entity (2), to calculate the Hamming weight of an error vector e = z ® aX ® bY, then, for the verifier entity (2), to accept the authentication if the weights Hamming of the error vectors e verify a relation of comparison to a parameter {T, t) function of the probability η. 2. The method according to claim 1, wherein said comparison relation consists in that the sum of the Hamming weights of the error vectors e obtained on the revolutions is less than a parameter equal to a threshold T. 3. The method of claim 2, wherein the threshold T is r (η + ε) m where ε is a margin less than 1/2. The method of claim 1, wherein said comparison relation is that the Hamming weight of the error vector e obtained at each turn is less than a parameter equal to a threshold t. The method of claim 1, wherein said comparison relation is that the Hamming weight of the error vector e obtained at each turn is equal to a parameter equal to a value t. 6. Method according to one of claims 4 or 5, wherein t is (η + ε) m where ε is a margin less than 1/2. The method of any one of claims 1 to 6, wherein said X and Y matrices are Toeplitz matrices. 8. Entity intended to be authenticated by a checking entity (2), said entities sharing a pair of secret keys X and Y, characterized in that said entity (1) to be authenticated comprises means (10) for storing secret keys X and Y constituted by binary matrices nxm (n, m> 1), means (12) for communication with the auditing entity (2), and calculation means (11) able to perform r times (r> 1) the steps of: drawing lots and transmitting to the checking entity (2) a binary vector b of n bits, to receive from the checking entity (2) a binary vector with n bits, randomly drawing a binary noise vector c of m bits, each of said m bits being equal to 1 with a probability η less than 1/2, and calculating and transmitting to the auditing entity (2) a response vector z of m bits equal to z = aX ® bY θ c. 9. A computer program comprising program instructions for implementing the steps of claim 8 when said program is executed by a computer belonging to said means (1 1) for calculating the entity (1) to be authenticated. 10. Verifying entity sharing a pair of secret keys X and Y with an entity (1) to be authenticated, characterized in that said checking entity (2) comprises means (20) for storing secret keys X and Y constituted by matrices nxm binaries (n, m> l), means (22) for communication with the entity (1) to be authenticated, and means (21) for calculating able to carry out r times (r> 1) the steps of: to receive from the entity (1) to authenticate a binary vector b of n bits, to draw lots and to transmit to the entity (1) to authenticate a binary vector a of n bits, to receive from the entity (1) to authenticate a response vector z of m bits, - to calculate the Hamming weight of an error vector e = z ® aX ® bY, and to accept the authentication if the Hamming weights of the r error vectors e verify a comparison relation to a parameter (T, t) function of a predetermined probability η. A computer program comprising program instructions for carrying out the steps of claim 10 when said program is executed by a computer forming part of said checking entity (2) computing means (21).